Question

Solve this Integeral :​

Answer 1

Question :

Integrate

$\sf\int\limits_{0}^1\tan{}^{-1}(\frac{1}{x^2- x +1})$

Solution :

$\sf\int\limits_{0}^1\tan{}^{-1}(\frac{1}{x^2- x +1})$

$\sf=\int\limits_{0}^1\tan{}^{-1}[\frac{x+(1-x)}{1-x(x-1)}]$

We know that

$\sf\tan {}^{-1} x + \tan {}^{ - 1} y = \tan {}^{ - 1} ( \dfrac{x + y}{1 - xy})$

Then ,

$\sf=\int\limits_{0}^1\tan{}^{-1}x+\int\limits_{0}^1\tan{}^{-1}(1-x)$

Now , Solve Further , by integration of parts .

Then ,

$\sf\int\limits_{0}^1\tan{}^{-1}[\dfrac{1}{x^2- x +1}]=\dfrac{\pi}{2}-\log(2)$

Note :

For step by step explanation

refer to the attachment .